Motion in a circle

Question 1

Define

radian

Answer 1

The angle subtended at the centre of a circle by an arc of length equal to the radius

x radians = x degrees x (π/180)

Question 2

Define

angular displacement

Answer 2

The angle of the arc through which an object has moved from its original position (measured in radians)

Question 3

Define

angular velocity

Answer 3

The rate of change of the angular displacement of an object as it moves along a curved path

ω = Δθ/Δt (rad s-1)

for an object moving along a complete circle, where T is the time taken for one complete round:

ω = 2π/T = 2πf

Question 4

How do you calculate the speed of an object moving along a curved path?

Answer 4

v = rω

Question 5

What causes circular motion?

Answer 5

• objects travelling in a circle can have steady speed but changing velocity
• this means they have acceleration
• according to Newton’s first law, this means there MUST be an unbalanced force acting upon the object
• this force is the centripetal force, and it acts perpendicularly to the direction of motion, towards the centre of the circle
• this force causes circular motion

Question 6

Define

centripetal force

Answer 6

The net force acting on an object moving in a circle; it is always directed towards the centre of the circle

F = mv²/r = mrω²

Question 7

How do you calculate centripetal acceleration?

Answer 7

a = v/r² = rω²

Question 8

How can a conical pendulum be used to determine centripetal force?

Answer 8

1. use an object of known mass
2. work out the angle θ
   
   1. put a screen behind with a torch
   2. spin the pendulum using a motor
   3. mark the shadow and measure the angle
3. use a force diagram to calculate the horizontal component of the tension in the string - this is the centripetal force

Question 9

For each of the following, state the origin of the centripetal force:

1. Airplane banking
2. Car turning on flat road
3. Car turning on banked road
4. Going over a bump

Answer 9

1. Airplane: lift
2. Car on flat road: friction
3. Car on banked road: weight and friction
4. Going over a bump: weight

Question 10

Describe motion in a vertical circle of a mass at the end of a string

Answer 10

F_c is constant at all points but the tension in the string changes according to the position of the mass

1. At the bottom of the circle: F_c = T₁ - mg
2. At the top of the circle: F_c = T₂ + mg
3. At the sides of the circle: F_c = T₃